The Number Column Puzzle

When does a 'creative' solution become 'cheating'?

02 December 2019

The numbers 1 to 9 have been written on cards and placed into two columns, like this:

                                 4                1

                                 5                2

                                 6                3

                                 9                7

                                                   8

Here's the puzzle: move just one card so that the two columns add to the same total.

Have a think about it before you read the solution below- and if you already know a solution, try to find a SECOND one.  Remember, the only instruction is that you are allowed to move one card.

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The column on the left adds to 24 and the one on the right adds to 21.  Any attempt to match the totals by moving one card to the other column proves fruitless, and there is a good reason for this.  The nine cards add to 45, so if the two columns are to be the same, both columns have to add to 22 1/2 , which isn't possible.

One idea is to get rid of the difference of 3 by removing the 3 card - only to realise that annoyingly the 3 is in the wrong column for this to work.

At which point many solvers get stuck.  But there is usually one who has an Aha! moment. Why not turn the 9 card upside down so it becomes a 6.  It works, both totals are now 21.  At this point, reactions differ. Some people say "Aaaah", while others say "Doh!".  When I present this puzzle to maths teachers, I ask them whether they think this solution is 'creative' or 'cheating'.  Typically the vote is split about 75-25 in favour of it being creative.  The rest think it's unfair - though they agree that the solution obeys the rules, albeit not the rules they were assuming would apply.

And this is an uncomfortable truth about any solution that pushes the conventional boundaries. Some will see it as creative, and others will think it is cheating.   The boundary between creativity and cheating depends on what you believe the unwritten rules are. 

This is something that differentiates a puzzle from a maths question. If an exam board posed this puzzle, most students would deem the solution unfair because this sort of trick doesn't follow the generally accepted conventions of how maths problems should be solved.  (The trick in this example is a different interpretation of the word 'move'.)

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When I first showed this puzzle to my own children a few years ago, I was interested to see how long it would be before they thought of inverting the 9.  But instead, struggling to find a solution, my eldest picked up the '1' card and tentatively placed it on top of the '5' card in the other column - and spotted that the two columns now added to 20.  This solution had never occurred to me, but it seemed as fair as the flipped 9.  The following day, I tweeted the puzzle to see if there were any other solutions lurking out there.  By lunchtime I'd received more than ten solutions.  Some of them were variants of picking up one card and putting it on another.  Others had added a twist of taking one card and turning it over so its blank back was showing, and then putting it on top of another card so that both numbers were hidden.  For example take the '5' card, turn it over and place it on the '2' card, so that both columns now add to 19.

One person suggested putting the 2 at the bottom left of the 3 to make it appear to be 2^3 (= 8), so both columns now add to 24.  Is that creative or cheating?  I like it, but if this had been the only possible solution it might have felt like a groaner.

And then there was somebody who suggested just picking up one card and putting it back in the same place.  "One card has moved, and the totals are the same (as they were at the start)".  I'm afraid that one definitely crosses my boundary.